Some Observations on Dyson's New Symmetries of Partitions

نویسندگان

  • Alexander Berkovich
  • Frank G. Garvan
چکیده

We utilize Dyson’s concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler’s Pentagonal Number Theorem. We streamline Dyson’s bijection relating partitions with crank ≤ k and those with k in the Rank-Set of partitions. Also, we extend Dyson’s adjoint of a partition to MacMahon’s “modular” partitions with modulus 2. This way we find a new combinatorial proof of Gauss’s famous identity. We give a direct combinatorial proof that for n > 1 the partitions of n with crank k are equinumerous with partitions of n with crank −k.

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عنوان ژورنال:
  • J. Comb. Theory, Ser. A

دوره 100  شماره 

صفحات  -

تاریخ انتشار 2002